A231691 Cardinalities of the symmetric operad of dotted red and white trees.

1, 6, 74, 1476, 41032, 1464672, 63865328, 3290120832, 195537380704, 13169097667584, 991181618539136, 82450282595311104, 7511417235983147008, 743790032122343820288, 79541198937597284060672, 9136079502141558495310848, 1121720442822518015112749056, 146607501639123412303738884096, 20322509742114322789584125210624, 2978025324234142178848508363882496

Offset

1,2

Links

Gheorghe Coserea, Table of n, a(n) for n = 1..300

F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.

Formula

E.g.f. A(x) satisfies -A(x) - g(-A(x)) = x where g is the E.g.f. of A052878. - Gheorghe Coserea, Jan 18 2017, edited by Robert Israel, Sep 27 2018

a(n) ~ sqrt((5 + 7*s + 3*s^2) / (7 + 13*s + 5*s^2)) * n^(n-1) / ((log((1+3*s+s^2)/(1+s))-s)^(n - 1/2) * exp(n)), where s = A060006 - 1 = -1 + (27/2 - 3*sqrt(69)/2)^(1/3)/3 + ((9 + sqrt(69))/2)^(1/3)/3^(2/3). - Vaclav Kotesovec, Apr 21 2020

Example

A(x) = x + 6*x^2/2! + 74*x^3/3! + 1476*x^4/4! + 41032*x^5/5! + ...

Maple

S:= series(RootOf(y=-x-ln((1+x)/(1+3*x+x^2)), x), y, 21):
seq(coeff(S, y, n)*n!, n=1..21); # Robert Israel, Sep 27 2018

Mathematica

terms = 20; (CoefficientList[InverseSeries[Log[x^2 + 3x + 1] - Log[1+x] - x + O[x]^(terms+1)], x]*Range[0, terms]!) // Rest (* Jean-François Alcover, Sep 16 2018, after Gheorghe Coserea *)

Prog

(PARI)
N=21; x = 'x + O('x^N); Vec(serlaplace(serreverse(log(x^2+3*x+1) - log(1+x) - x))) \\ Gheorghe Coserea, Jan 18 2017

Crossrefs

Cf. A052878, A231690.

Sequence in context: A057783 A317344 A357011 * A177561 A269334 A069852

Adjacent sequences: A231688 A231689 A231690 * A231692 A231693 A231694

Keyword

nonn

Author

N. J. A. Sloane, Nov 14 2013

Extensions

Offset changed and more terms from Gheorghe Coserea, Jan 15 2017

Status

approved